A370258 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(12 + 24\,x + 12\,x^{2} + 14\,n + 28\,n x + 14\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(12 + 68\,x + 72\,x^{2} - 12\,x^{3} - 28\,x^{4} + 14\,n + 78\,n x + 69\,n x^{2} - 40\,n x^{3} - 45\,n x^{4} + 4\,n^{2} + 22\,n^{2} x + 15\,n^{2} x^{2} - 20\,n^{2} x^{3} - 17\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(13\,x + 67\,x^{2} + 123\,x^{3} + 97\,x^{4} + 28\,x^{5} + 8\,n x + 41\,n x^{2} + 75\,n x^{3} + 59\,n x^{4} + 17\,n x^{5}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 12\,x + 26\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	12	26								39
1	1	24	129	190							344
2	1	40	384	1312	1462						3199
3	1	60	890	5180	12930	11584					30645
4	1	84	1770	15360	63585	124884	93536				299220
5	1	112	3171	38010	229635	735084	1189433	765314			2960760
6	1	144	5264	82880	679560	3129280	8152480	11212800	6323270		29585679
7	1	180	8244	164472	1745352	10775520	39965520	87703200	104876190	52638760	297877439

All 15 rows

Technical details

Node id: n-a7f4b72a0d518b5f · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-c4c2d2a9fc3745ce

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1,  12,    26;
  1: 1,  24,   129,    190;
  2: 1,  40,   384,   1312,    1462;
  3: 1,  60,   890,   5180,   12930,    11584;
  4: 1,  84,  1770,  15360,   63585,   124884,     93536;
  5: 1, 112,  3171,  38010,  229635,   735084,   1189433,     765314;
  6: 1, 144,  5264,  82880,  679560,  3129280,   8152480,   11212800,    6323270;
  7: 1, 180,  8244, 164472, 1745352, 10775520,  39965520,   87703200,  104876190,   52638760;
  8: 1, 220, 12330, 303360, 4028010, 31849272, 156956520,  486511680,  921766410,  974868760,  440815036;
  9: 1, 264, 17765, 527670, 8546670, 83786472, 524669838, 2145989340, 5707676370, 9511131520, 9016432986, 3709445084;

%F Formula

The row polynomials satisfy (4*n^2+14*n+12+(8*n^2+28*n+24)*x+(4*n^2+14*n+12)*x^2)*P(n,x) = (4*n^2+14*n+12+(22*n^2+78*n+68)*x+(15*n^2+69*n+72)*x^2-(20*n^2+40*n+12)*x^3-(17*n^2+45*n+28)*x^4)*P(n-1,x) + ((8*n+13)*x+(41*n+67)*x^2+(75*n+123)*x^3+(59*n+97)*x^4+(17*n+28)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 12*x + 26*x^2.

%t Mathematica

P[0, x_] := 1 + 12*x + 26*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 14*n + 12 + (22*n^2 + 78*n + 68)*x + (15*n^2 + 69*n + 72)*x^2 - (20*n^2 + 40*n + 12)*x^3 - (17*n^2 + 45*n + 28)*x^4)*P[n - 1, x] + ((8*n + 13)*x + (41*n + 67)*x^2 + (75*n + 123)*x^3 + (59*n + 97)*x^4 + (17*n + 28)*x^5)*D[P[n - 1, x], x])/(4*n^2 + 14*n + 12 + (8*n^2 + 28*n + 24)*x + (4*n^2 + 14*n + 12)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 12*x + 26*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 14*n + 12 + (22*n^2 + 78*n + 68)*x + (15*n^2 + 69*n + 72)*x^2 - (20*n^2 + 40*n + 12)*x^3 - (17*n^2 + 45*n + 28)*x^4)*v[i - 1] + ((8*n + 13)*x + (41*n + 67)*x^2 + (75*n + 123)*x^3 + (59*n + 97)*x^4 + (17*n + 28)*x^5)*deriv(v[i - 1], x))/(4*n^2 + 14*n + 12 + (8*n^2 + 28*n + 24)*x + (4*n^2 + 14*n + 12)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n, k) = binomial(n, k)*binomial(2*n+k, k), 0 <= k <= n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.